Heat flows between two bodies in contact until the system attains thermal equilibrium. What happens if the bodies in contact having different specific heat capacity and same temperature. Does the flow of heat occurs until the heat energy between the bodies become equal since the amount of heat need to rais the temperature is different?
[Physics] Can heat transfer occur between two bodies with same temperature and different specific heat capacity
thermodynamics
Related Solutions
https://en.wikipedia.org/wiki/Standard_molar_entropy?wprov=sfti1
$$dQ = T \ dS \tag1$$ $$dQ = C \ dT \tag2$$
Interesting, right? In $(1)$, the whole $T$ multiplies the infinitesimal $\frac{\text{J}}{\text{K}}$. In $(2)$ it's the opposite: the whole $\frac{\text{J}}{\text{K}}$ multiplies the infinitesimal $T$.
But you hinted that you knew that yourself already. Let's cut to the chase: both are different beasts entirely, just like heat and torque are not related just because they carry the same unit (joules are newton-meters, right?).
However, if you still want a defining difference between them, other than "they're just different", I'd give you this:
Entropy by itself is not useful and cannot even be measured. What is useful are changes in entropy, or how it differs from one state to the other. In this sense, it's akin to internal energy and enthalpy, for which only relative values matter. Heat capacity, on the other hand, can have its absolute value determined experimentally, and it won't depend on a reference value like entropy does. Its absolute value is immediately useful, if you will. In this sense, it's akin to pressure and specific volume, for which absolute values make sense.
I'll answer your second question first, because then your first one is easier. In short, yes, the equilibration of temperature between two bodies is absolutely universal - it doesn't depend on how the heat is transferred, and in particular it does apply to radiative transfer. And, indeed, once two bodies have reached thermal equilibrium through radiative transfer, the radiation in between them has a temperature that is the same as the temperature of the two bodies.
Let us now imagine two bodies coming into radiative equilibrium. We'll say that one of them is a hollow sphere and the other one is inside it, because then we don't have to consider the surrounding environment (which I'll get to shortly). The inner body will be giving off heat at a rate proportional to its temperature to the fourth power, and this doesn't depend on the temperature of its surroundings at all. But it's also absorbing heat, at a rate that does depend on the temperature of its surroundings. (It's proportional to the fourth power of the outer body's temperature.) So when they come into equilibrium, the inner body is giving off and receiving heat at exactly the same rate. Radiation is still occurring, but heat flow is not, because the radiation coming in cancels the radiation going out, so there's no net flow of energy. This answers your first question.
Often we don't consider this because we're thinking about something that's a lot hotter than its surroundings. Because $T^4$ increases very rapidly with $T$, the radiative energy transfer from the environment is often small enough to be ignored in this case. In particular, for a body in space that's not exposed to sunlight, the relevant incoming radiation is the cosmic microwave background, which has a temperature of only 3 kelvin and can be ignored for most purposes.
Best Answer
May be an analogy will help. If two tanks containing water are connected by a pipe, water flows from the tank with higher level to that with lower level of water. Temperature is analogous to water level. Heat capacity is analogous to floor area of the tank. The tank with a larger floor area can hold greater amount of water (for a given water level), but that has nothing to with determining whether and in which direction water will flow.